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An introduction to diffusion

- Thermochron, Fall 2005

TEXTS

- Heat conduction - Carslaw and Jaegger
- Diffusion - Crank, 1975

Diffusion is analogous to heat conduction

- The rate of transfer of diffusing substance

through unit area of a section is proportional to

the concentration gradient measured normal to the

section (Ficks law, 1855).

Dashed for heat sources and sinks

Similar to heat transfer- same thing, really

- Rate of transfer of heat per unit area is

proportional to the thermal gradient (Fouriers

Law of heat conduction, 1822). - One deals with the diffusion of heat the other

with the diffusion of mass.

Diffusion coefficient

QUESTIONS?? What are the units for D? Why the

negative sign for both forms of diffusion?

Answers

- The minus sign reflects the fact that diffusion

takes place in the direction opposite to

increasing concentration (or heat) - Units are area/time, e.g. cm2/sec.
- These simple formulations apply only to perfectly

isotropic mediums.

Deriving the fundamental equation for diffusion

in an isotropic medium

Requires applying the law of conservation of

energy to a volume, e.g. a parallelepiped of

lengths 2x, 2y, and 2z. EinEgeneratedEstoredEo

ut

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The equivalent equation for heat conduction

Some implications

- The changes in concentration due to diffusion

over a characteristic time interval t will

propagate a distance on the order of (Dt)1/2 - Similar equation for conduction
- This simplification x (Dt)1/2 gives us a mean to

determine back of the envelope diffusion scales. - If the length scale is x (grain size), how long

will it take for diffusion to operate at that

length scale?

Note As you will see soon, diffusion is a

thermally activate process and the assumption of

constant D is a very simplifying one. However, D

is typically constant over a given small range of

temperatures.

Example

- Upper mantle minerals have grain sizes of mm

length scale. Lets say 1 mm on average. - The diffusion coefficient for most elements in

olivines and pyroxenes (at T 1000 0C) is on the

order of 1 x 10-15 cm2/sec. - What is the time scale of diffusional

equilibration of a mantle rock and what is the

prospect of using these rocks for geochronology

of mantle events?

Answers

- Takes less than 1 My (in this case 0.3 Ma) to

erase any diffusional gradients - Prospects of mantle rocks to preserve any kind of

diffusional gradients and thus age info not

good to zero.

Same applies to heat conduction

- Similar to x (Dt)1/2 , one can apply x (Kt)1/2

to heat conduction - E.g. what is the time scale for a granite body of

size x to cool?

r

cold

1 km body, 10 km, 100 km body

hot

Note the non-linearity

- Does a magma body ten times larger cool ten time

slower?

100 times slower !

Goals

- Fundamentally, we are in pursuit of diffusion

laws and the solutions to those laws because they

constrain the effective temperatures below which

the effective transport of elements and isotopes

of interest stops. - That is when the isotopic clock starts and

quantifying the conditions under which that

happens is equally important as the decay process

itself.

A road map for the next three lectures

- Find simple analogies to exemplify diffusion of

mass - Try to work out analytical solutions to super

simplified cases (sphere, plane geometries) - Bring in the math baggage associated with this
- Apply these solutions to the classic formulation

of closure temperature (Dodson, 1973) - Modern variations on the closure temperature

theory

An example

- Imaginary experiment pool of water and drop a

chlorine tablet. The pool is an infinite

reservoir that has no chlorine to start with. - What will be the distribution of Cl as a function

of x and t in the pool?????

Set initial and boundary conditions

CtabletC0 Cpool0 t0

Differentiating, one obtains the following

solution

If the concentration is originally located in a

point source and the medium is a plane

Which is the classic, analytical solution to a

point source and works ok for your pool although

the chlorine tablet is not a infinitely small and

the pool is not a infinitely large sheet..

Graphically..

- The problem is symmetric with respect to x
- The Pi-form of the constant is determined by the

fact that the chlorine diffuses not just in the

directions x and -x, but radially around the

tablet.

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Analogy

- A garnet immersed in an infinitely large

reservoir of biotite

Case 2 two long metal bars placed in contact end

to end.

- Another classic example in the diffusion book.
- Has an analytical solution
- Exemplifies the general form of solutions to more

complicated problems

Two infinite half spaces

Solution Consider that the half space is

composed of an infinite number of point sources

(our previous problem). Because of that, one can

obtain an analytical solution by superposing the

infinite of point sources. Overall, in math,

this sumation of linear solutions leads to an

exponential distribution of the solution.

Looks like the solution is of the form of a

well-known mathematical function called the error

function (erf)

Solutions to erf are available in tabulated form

or as an excel add-in etc etc.

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Solution

This is very similar to the heat conduction

solution for equivalent boundary / initial

conditions which the temperature at the interface

stays ct as the average between the two ts.

applies to dike intrusions etc

Solution

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Case 3 plane source of limited extent -

- A particular case of 2 with the source is limited

from -hltxgth - Integration is thus from x- h to xh instead of x

to infinity

Solution

Some concluding remarks from this intro lecture

- Mass diffusion is (almost) identical in its

treatment with heat conduction - aka heat

diffusion - Simplifying calculations can give us order of mag

info on length and time scales of diffusion - Analytical solutions are found for some of the

simplest initial and boundary conditions - The most common type of solution of these

problems involves the error function - More complicated diffusion problems do not have

analytical solutions discussions so far focused

on constant D and isotropic 1D examples.